Electronic structure of correlated electron systems

Lecture 2

Band Structure approach vs atomic

Band structure

• Delocalized Bloch states

• Fill up states with electrons starting from the lowest energy

• No correlation in the wave function describing the system of many electrons

• Atomic physics is there only on a mean field like level

• Single Slater determinant states

Atomic

• Local atomic coulomb and exchange integrals are central

• Hunds rules for the Ground state -Maximize total spin-Maximize total angular momentum-total angular momentum J =L-S<1/2 filled shell , J=L+S for >1/2 filled shell

• Mostly magnetic ground states

Taken from the lecture notes of IlyaElfimov UBC which will also be put on

a web site.

DFT and band theory of solidsThe many electron wave function is assumed to be a

single Slater determinant of one electron Bloch states commensurate with the periodic symmetry of the atoms in the lattice and so has no correlation in it

kNkkkN

321!

1The single particle wave functions φcontain the other quantumnumbers like atomic nlm and spin. k represents the momentum vector

The effects of correlation are only in the effectiveone particle Hamiltonian

xcnucleareff vrdrr

rnvv

'

'

)'( 3

Using such a single Slater determinant means that there is no correlation in this many electron wave function. The effects of correlation are in H only in the form of effective single particle potentials. In a simple physical picture of this we imagine that each electron repels other electrons and produces a suppression of density around it. This is referred to as an “ exchange correlation hole”. In this form of DFT the effect of this is in the potential but cannot be included in the many electron wave function

Abinito approach to DFT as applied to band theory of solids

• We can get the “exact” ground state energy and ground state density (crystal structure ) provided the exchange correlation potential is known

• Recall that the ground state has few properties i.e.energy and density, It is the excited states that determine the response to External perturbations such as fields

• If we had the ground state wave function on the other hand we could at least guess at some of the properties such as metallic or insulating, magnetic or not etc.

DFT band theory in LDA

Remember that the one electron wavefunctionsϕ in the above have no physical meaning in fact and neither do the one electron energies . They are merely a tool to calculate the total ground state energy and density.

Note also that the total many body wave function also has no meaning physically and actually is not an eigenfunction of the original Hamiltonian.

Non the less in band theory these are taken to be the “quasi” particle energies and wave functions in comparisons with experiment.

dft

G

dft

exact E

n

i ji

jijiiexactrr

erWH

1

2

,

2 2/12/1

dft

G

dft

dft E

The DFT wave function is not an eigen function of theexact Hamiltonian

However

The single particle wave functions have k as a good quantum number in DFT. The electron -electron term in H exact willalways have scattering matrix elements

'''''' kkkk i.e. 2 electrons k,k’ scatter from below to above k’’,k’’’ the Fermi Energy.

iRik

i

ik eRrN

)(/1

ji rr

e

2 These matrix elements will be largest if the two electrons are on the same atomic site in whichcase = U i.e. Hubbard.

. Consider tight binding one electron Wave functions

Now look at the electron electronscattering due to a Hubbard like U

N

UH

kk

kkgdft

'''''

'int

With k+k’ -k’’-k’’’ = 0 for momentum conservationWhere k and k’ are occupied states and k’ and k’’’are unoccupied.

GOES TO ZERO FOR N INFINITE. However we have to sum over allthese scattering events and if U is comparable to the band widthW or the Fermi energy as measured from the bottom of the bandthen basically all electrons are involved and in the total we haveto sum over these resulting in an effective a scattering matrix element of one electron due to interaction with all the others of U. This demonstrates how one can be misled by looking at asingle off diagonal matrix element and that for U comparableto W the effective scattering is actually U. IN OTHER WORDS

ψdft is far from being an eigenfunction of the exact Hamiltonian

if U is comparable to W.

For systems with R<<D

For R<<D and Large U we get a qualitatively different ansatz wave function. Consider a half filled s like band i.e. 1 electron per atom on average. For a zero band width W the ground state wave function might look more like .

NAN

321!

1Where the integers now label sitesAnd the one electron wave functionsAre atomic orbitals center at cite i

The off diagonal matrix elements now involve W i.e. hopingso this is better for U>>W. Note that also here we have on the Average one electron per atom as in the DFT wave function

In the zero band width limit and again a single s like band the electron charge density in DFT would also correspondto exactly one electron per atom but the wave function would be a single Slater determinant of one electron Bloch wavesand not a single Slater determinant of atomic site localizeds orbitals with one electron at each site. In the DFT casethere would be two electrons with opposite spin in each k statewhile in the atomic case each atom would have one unpaired electron and S=1/2. This would yield a paramagnetic susceptibilityi.e. 1/Temp as expected for a collection of independent atomswith one electron per atom. In the DFT approach the material would be expected to be metallic since the band is half full andthe band is centered at Ef . The atomic like approach would obviously yield an insulator since to move an electron we reguirean excitation to an doubly occupied cite resulting in an excitation Energy of U>>W.

Including the exchange correlation hole in DFT wave function

In our simple example of a lattice of H atoms with one electron per atom we could conceptually include an exchange correlation hole in the wave function. Basically it would correspond to every occupied by one electron is also forced to have an unoccupied opposite spin orbital on that atom. This if we could impliment it would result in qualitatively the same properties as the atomic limit. However we don’t really know how to impliment this.

Configuration interaction approachThe one electron wave functions in ψ atomic do

not possess the symmetry of the lattice which in chemistry is called a broken symmetry ansatz. To include intersite hoping perturbatively we consider mixing in electron configurations with now empty sites and others with two electrons on a site.

t

Energy =UT=nn hoping integral

Mixing in of this excited state wave function amplitude = t/U But there are an infinite Number of these virtual excitations in a configuration interaction approach.

Energy

Ef

Density of States

R>>D band not full and not emptyResults in a narrow band at Ef

R<D

Band theory result for a coexistence of extreme states

Huge successes of DFT• Obtain the correct ground state crystal structure and

quite accurate lattice parameters for a large diversity of systems

• Obtain the correct magnetic structure for a large diversity of materials

• First principles method to calculate electron phonon coupling by introducing lattice distortions and obtaining the new ground state energy

• Extremely important role in also correlated electron systems for the determination of parameters to be used in many body Hamiltonian approaches.

What do we really mean by states above and below Ef?

• E<Ef eigenstates of the N-1 electron system i.e. ionization states of the N electron system i.e. Reachable by photoemission

• E>Ef eigenstates of the N+1 electron system i.e. electron affinity states of N particle system i.e. reachable by inverse photo electron spectroscopy states

• These two differ by two electrons

Experimental measurement

• States below Ef ----Photoelectron spectroscopy PES or ARPES. Removes one electron

• States above Ef ---Inverse photoelectron spectroscopy IPES. Adds one electron

Angular resolved photo and inverse photo electron spectroscopy

• Consider a single crystal

• Eigenstates have periodicity of the crystal

• Momentum (k) is a good quantum number

• Photon energy is low, long wave length, zero momentum change

• Energy conservation E(K)=e(k) +e(photon) , e(k) = band dispersion

• Momentum conservation K = k + G

One example of the many N-1 electron eigenstates which in thiscase result in Various numbers of vibrational excitations

The electrons in H2 are “ dressed” with lattice molecular vibrations-phonons in solids

Electron removal spectral function for H2 molecular solid with large lattice spacing

• For large lattice spacing band width is zero.

• Sudden removal of an electron decreases the binding between the H atoms in a molecule causing an increase in the equilibrium bond length for He+ molecule.

• The N-1 electron eigenstates will involve the vibrational excitations of the H2+ molecule

• Energy conservation will result in the photoelectron exhibiting peaks including the possible vibrational excitations of H2+

• The intensity of each peak will be given by the overlap integral of the ground state H2 vibrational wave function with that of the vibrationaleigenstates of the H2+ molecule.

• The lowest energy N-1 electron state will have a spectral weight of about 10% of the total which in a solid would mean a quasi particle weight of 10 reducing the band width by 10.

• The electron in the solid is dressed by Phonons resulting in a polaronic like state

Andrea Damascelli will give a lecture in beginning of March I hope about angular resolved photoelectron.

Mona Berciu has agreed to give a lecture in the beginning of March as an introduction to electron phonon coupling in solids.

ARPES Cu

3d bands

4s,4p,band

Cu is d10 so one d holeHas no other d holes to Correlate with so 1 part.Theory works if the only Important interaction is The d-d interaction.Great agreement with DFT

Points –exp.Lines - DFT

Angular resolved photoelectron spectroscopy (ARPES) of Cu metal Thiry et al 1979

What if we remove 2- d electrons ?Two hole state with Auger spectroscopy

3d

2p 932eV

PhotonPhotoelectronAuger electron

E(photon)-E(photoelectr) = E(2p) , E (2-d holes)= E(2p)-E(3d)-E(Auger)

U = E( 2-d holes) -2xE(1-d hole)

Example is for Cu withA fully occupied 3d band

Auger spectroscopy of Cu metalAtomic multipletsLooks like gas phase

U>W

Hund’s ruleTriplet F is Lowest

Antonides et al 1977 Sawatzky theory 1977

The L3M45M45 Auger spectrum of Cu metal i.e final state has 2 -3d holes on the Atom that started with a 2p hole. Solid line is the experiment. Dashed line is one Electron DFT theory, vertical bars and lables are the free atom multiplets for 8- 3d electrons on a Cu atom . Ef designates the postion of the Fermi level in the DFT .

Two hole bound states

We note that for Cu metal with a full 3d band in the ground state one particle theory works well to describe the one electron removal spectrum as in photoelectron spectroscopy this is because a single d hole has no other d holes to correlated with. So even if the on site d-d coulomb repulsion is very large there is no phase space for correlation.

The strength of the d-d coulomb interaction is evident if we look at the Auger spectrum which probes the states of the system if two electrons are removed from the same atom

If the d band had not been full as in Ni metal we would have noticed the effect of d-d coulomb interaction already in the photoemission spectrum as we will see.

D shells are complicated by multipletstructure

• Atomic physics – d orbital is 5 fold degenerate not including the spin and neglecting the spin orbit coupling .

• Two d electrons or holes with orbital angular momentum =2 and spin of ½ can couple into total angular momentum states L with total spin 1 or 0 as follows ; singlet S, singlet G, singlet D and triplet P and triplet F

• The energy separations in the Cu Auger spectrum are from atomic coulomb integrals with triplet F as the lowest energy state for 8 d electrons as given by Hunds’ rule

For U>>W and in the presence of unfilled bands the one particle removal spectrum will be very different

from that of a filled band

Compare the PES of Cu metal with a full d band to that of Ni with on the

average 0.6 holes in the 3d band

Phtoemission from Full versus partly full 3d bands

• If the band is full as in Cu the removal spectrum of one electron leaves only one hole and one particle theory will work. This is why DFT band theory works so well for Cu.

• In Ni metal the 3d band is partly full as seen below. In this case things are more difficult since the atomic 3d occupation number fluctuates ( quantum fluctuations) resulting from the band structure.

Lower fig shows the Cu Density of states as measured in angular Integrated photoemission spectroscopy

Density ofStates or Photoemission Intensity. The broadBand crossing EfIs the Cu 4s “ free Electron like band

Contrast Cu with Ni • For Ni the 3d band crosses Ef. 3d states are in the

limit R<D i.e. correlation if bands are not full or empty on average about 9.4 per atom.

• Snapshot picture of the local atomic d occupation

• d9 d10 d9 d9 d10 d10 d9 --------

• Removing a d electron yields states like d8 which involves U and d9 which does not

• We can expect two energy regions for d electron removal from Ni separated by about U in energy

DFT band theory density Of states

Photoemission spectrum of Ni metal Depends on phtotonenergy

Ni metal 3d density of states and phtoemissionspectrum

3d9

3d10

A Slater determinant of one electron Bloch states involves huge polarity fluctuations

• In band theory the probability that an atom has n d electrons ( d states are 10 fold degenerate) is purely statistical i.e. not taking into account the huge energy differences with n due to the interactions.

• In real life of course fluctuations from the average cost coulomb energy so in actual fact the d occupation will be much more peaked about the average. This is why we consider Ni to have either d10 or d9 and not higher deviations from the average occupation i.e. U is large

nmm cc

nnm

mnP

1

!!

!)(

Where m=degeneracy, c is the Electron concentration =9.4 for Ni

Lang Baer and Cox J Phys F 11, 121 (1981)

• Photoemission and inverse photoemission of all the rare earth metals

• Demonstrates the atomic multiplets of the 4f electron removal and addition states

• Intensities given by atomic coefficients of fractional parentage starting from the Hunds’ rule ground state

U = 12 eV

MORE ON RARE EARTHS

• The Hubbard U; as clearly demonstrated, its definition depends on which multiplets you take and depends strongly on the element. Convention is to either take the multipletaverage or the Slater F0 integral.

• The multiplet splitting is very close to the atomic value little SCREENING OF THE HUNDS RULES INTERACTIONS I.E. SLATER F2,F4,F6 INTERACTIONS

. We will come back to this later.

Note the atomic physics needed to describe the rare earth 4f electron

removal and addition spectrum

For the 3d transition metal compounds things are a lot more subtle. In some

cases we need the atomic approaches and in others one particle theory

seems to work very well